.. _sec_nadaraya-watson:
注意力汇聚：Nadaraya-Watson 核回归
==================================
上节介绍了框架下的注意力机制的主要成分 :numref:`fig_qkv`\ ：
查询（自主提示）和键（非自主提示）之间的交互形成了注意力汇聚；
注意力汇聚有选择地聚合了值（感官输入）以生成最终的输出。
本节将介绍注意力汇聚的更多细节，
以便从宏观上了解注意力机制在实践中的运作方式。
具体来说，1964年提出的Nadaraya-Watson核回归模型
是一个简单但完整的例子，可以用于演示具有注意力机制的机器学习。
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from mxnet import autograd, gluon, np, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l
npx.set_np()
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import torch
from torch import nn
from d2l import torch as d2l
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import tensorflow as tf
from d2l import tensorflow as d2l
tf.random.set_seed(seed=1322)
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\diilbookstyleinputcell
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import warnings
from d2l import paddle as d2l
warnings.filterwarnings("ignore")
import paddle
from paddle import nn
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\diilbookstyleinputcell
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n_train = 50 # 训练样本数
x_train = np.sort(np.random.rand(n_train) * 5) # 排序后的训练样本
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[07:15:08] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU
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\diilbookstyleinputcell
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def f(x):
return 2 * np.sin(x) + x**0.8
y_train = f(x_train) + np.random.normal(0.0, 0.5, (n_train,)) # 训练样本的输出
x_test = np.arange(0, 5, 0.1) # 测试样本
y_truth = f(x_test) # 测试样本的真实输出
n_test = len(x_test) # 测试样本数
n_test
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50
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n_train = 50 # 训练样本数
x_train, _ = torch.sort(torch.rand(n_train) * 5) # 排序后的训练样本
def f(x):
return 2 * torch.sin(x) + x**0.8
y_train = f(x_train) + torch.normal(0.0, 0.5, (n_train,)) # 训练样本的输出
x_test = torch.arange(0, 5, 0.1) # 测试样本
y_truth = f(x_test) # 测试样本的真实输出
n_test = len(x_test) # 测试样本数
n_test
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50
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n_train = 50
x_train = tf.sort(tf.random.uniform(shape=(n_train,), maxval=5))
def f(x):
return 2 * tf.sin(x) + x**0.8
y_train = f(x_train) + tf.random.normal((n_train,), 0.0, 0.5) # 训练样本的输出
x_test = tf.range(0, 5, 0.1) # 测试样本
y_truth = f(x_test) # 测试样本的真实输出
n_test = len(x_test) # 测试样本数
n_test
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50
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n_train = 50 # 训练样本数
x_train = paddle.sort(paddle.rand([n_train]) * 5) # 排序后的训练样本
def f(x):
return 2 * paddle.sin(x) + x**0.8
y_train = f(x_train) + paddle.normal(0.0, 0.5, (n_train,)) # 训练样本的输出
x_test = paddle.arange(0, 5, 0.1, dtype='float32') # 测试样本
y_truth = f(x_test) # 测试样本的真实输出
n_test = len(x_test) # 测试样本数
n_test
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50
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y_hat = y_train.mean().repeat(n_test)
plot_kernel_reg(y_hat)
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\diilbookstyleinputcell
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y_hat = torch.repeat_interleave(y_train.mean(), n_test)
plot_kernel_reg(y_hat)
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\diilbookstyleinputcell
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y_hat = tf.repeat(tf.reduce_mean(y_train), repeats=n_test)
plot_kernel_reg(y_hat)
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\diilbookstyleinputcell
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y_hat = paddle.repeat_interleave(y_train.mean(), n_test)
plot_kernel_reg(y_hat)
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# X_repeat的形状:(n_test,n_train),
# 每一行都包含着相同的测试输入（例如：同样的查询）
X_repeat = x_test.repeat(n_train).reshape((-1, n_train))
# x_train包含着键。attention_weights的形状：(n_test,n_train),
# 每一行都包含着要在给定的每个查询的值（y_train）之间分配的注意力权重
attention_weights = npx.softmax(-(X_repeat - x_train)**2 / 2)
# y_hat的每个元素都是值的加权平均值，其中的权重是注意力权重
y_hat = np.dot(attention_weights, y_train)
plot_kernel_reg(y_hat)
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\diilbookstyleinputcell
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# X_repeat的形状:(n_test,n_train),
# 每一行都包含着相同的测试输入（例如：同样的查询）
X_repeat = x_test.repeat_interleave(n_train).reshape((-1, n_train))
# x_train包含着键。attention_weights的形状：(n_test,n_train),
# 每一行都包含着要在给定的每个查询的值（y_train）之间分配的注意力权重
attention_weights = nn.functional.softmax(-(X_repeat - x_train)**2 / 2, dim=1)
# y_hat的每个元素都是值的加权平均值，其中的权重是注意力权重
y_hat = torch.matmul(attention_weights, y_train)
plot_kernel_reg(y_hat)
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\diilbookstyleinputcell
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# X_repeat的形状:(n_test,n_train),
# 每一行都包含着相同的测试输入（例如：同样的查询）
X_repeat = tf.repeat(tf.expand_dims(x_train, axis=0), repeats=n_train, axis=0)
# x_train包含着键。attention_weights的形状：(n_test,n_train),
# 每一行都包含着要在给定的每个查询的值（y_train）之间分配的注意力权重
attention_weights = tf.nn.softmax(-(X_repeat - tf.expand_dims(x_train, axis=1))**2/2, axis=1)
# y_hat的每个元素都是值的加权平均值，其中的权重是注意力权重
y_hat = tf.matmul(attention_weights, tf.expand_dims(y_train, axis=1))
plot_kernel_reg(y_hat)
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# X_repeat的形状:(n_test,n_train),
# 每一行都包含着相同的测试输入（例如：同样的查询）
X_repeat = x_test.repeat_interleave(n_train).reshape((-1, n_train))
# x_train包含着键。attention_weights的形状：(n_test,n_train),
# 每一行都包含着要在给定的每个查询的值（y_train）之间分配的注意力权重
attention_weights = nn.functional.softmax(-(X_repeat - x_train)**2 / 2, axis=1)
# y_hat的每个元素都是值的加权平均值，其中的权重是注意力权重
y_hat = paddle.matmul(attention_weights, y_train)
plot_kernel_reg(y_hat)
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d2l.show_heatmaps(np.expand_dims(np.expand_dims(attention_weights, 0), 0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
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d2l.show_heatmaps(attention_weights.unsqueeze(0).unsqueeze(0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
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\diilbookstyleinputcell
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d2l.show_heatmaps(tf.expand_dims(
tf.expand_dims(attention_weights, axis=0), axis=0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
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\diilbookstyleinputcell
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d2l.show_heatmaps(attention_weights.unsqueeze(0).unsqueeze(0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
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X = np.ones((2, 1, 4))
Y = np.ones((2, 4, 6))
npx.batch_dot(X, Y).shape
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(2, 1, 6)
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X = torch.ones((2, 1, 4))
Y = torch.ones((2, 4, 6))
torch.bmm(X, Y).shape
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torch.Size([2, 1, 6])
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X = tf.ones((2, 1, 4))
Y = tf.ones((2, 4, 6))
tf.matmul(X, Y).shape
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:class: output
TensorShape([2, 1, 6])
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X = paddle.ones((2, 1, 4))
Y = paddle.ones((2, 4, 6))
paddle.bmm(X, Y).shape
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[2, 1, 6]
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weights = np.ones((2, 10)) * 0.1
values = np.arange(20).reshape((2, 10))
npx.batch_dot(np.expand_dims(weights, 1), np.expand_dims(values, -1))
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array([[[ 4.5]],
[[14.5]]])
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weights = torch.ones((2, 10)) * 0.1
values = torch.arange(20.0).reshape((2, 10))
torch.bmm(weights.unsqueeze(1), values.unsqueeze(-1))
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tensor([[[ 4.5000]],
[[14.5000]]])
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weights = tf.ones((2, 10)) * 0.1
values = tf.reshape(tf.range(20.0), shape = (2, 10))
tf.matmul(tf.expand_dims(weights, axis=1), tf.expand_dims(values, axis=-1)).numpy()
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array([[[ 4.5]],
[[14.5]]], dtype=float32)
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weights = paddle.ones((2, 10)) * 0.1
values = paddle.arange(20, dtype='float32').reshape((2, 10))
paddle.bmm(weights.unsqueeze(1), values.unsqueeze(-1))
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\diilbookstyleoutputcell
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:class: output
Tensor(shape=[2, 1, 1], dtype=float32, place=Place(cpu), stop_gradient=True,
[[[4.50000000 ]],
[[14.50000000]]])
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class NWKernelRegression(nn.Block):
def __init__(self, **kwargs):
super().__init__(**kwargs)
self.w = self.params.get('w', shape=(1,))
def forward(self, queries, keys, values):
# queries和attention_weights的形状为(查询数，“键－值”对数)
queries = queries.repeat(keys.shape[1]).reshape((-1, keys.shape[1]))
self.attention_weights = npx.softmax(
-((queries - keys) * self.w.data())**2 / 2)
# values的形状为(查询数，“键－值”对数)
return npx.batch_dot(np.expand_dims(self.attention_weights, 1),
np.expand_dims(values, -1)).reshape(-1)
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class NWKernelRegression(nn.Module):
def __init__(self, **kwargs):
super().__init__(**kwargs)
self.w = nn.Parameter(torch.rand((1,), requires_grad=True))
def forward(self, queries, keys, values):
# queries和attention_weights的形状为(查询个数，“键－值”对个数)
queries = queries.repeat_interleave(keys.shape[1]).reshape((-1, keys.shape[1]))
self.attention_weights = nn.functional.softmax(
-((queries - keys) * self.w)**2 / 2, dim=1)
# values的形状为(查询个数，“键－值”对个数)
return torch.bmm(self.attention_weights.unsqueeze(1),
values.unsqueeze(-1)).reshape(-1)
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class NWKernelRegression(tf.keras.layers.Layer):
def __init__(self, **kwargs):
super().__init__(**kwargs)
self.w = tf.Variable(initial_value=tf.random.uniform(shape=(1,)))
def call(self, queries, keys, values, **kwargs):
# 对于训练，“查询”是x_train。“键”是每个点的训练数据的距离。“值”为'y_train'。
# queries和attention_weights的形状为(查询个数，“键－值”对个数)
queries = tf.repeat(tf.expand_dims(queries, axis=1), repeats=keys.shape[1], axis=1)
self.attention_weights = tf.nn.softmax(-((queries - keys) * self.w)**2 /2, axis =1)
# values的形状为(查询个数，“键－值”对个数)
return tf.squeeze(tf.matmul(tf.expand_dims(self.attention_weights, axis=1), tf.expand_dims(values, axis=-1)))
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class NWKernelRegression(nn.Layer):
def __init__(self, **kwargs):
super().__init__(**kwargs)
self.w = paddle.create_parameter((1,), dtype='float32')
def forward(self, queries, keys, values):
# queries和attention_weights的形状为(查询个数，“键－值”对个数)
queries = queries.reshape((queries.shape[0], 1)) \
.tile([keys.shape[1]]) \
.reshape((-1, keys.shape[1]))
self.attention_weight = nn.functional.softmax(
-((queries - keys) * self.w)**2 / 2, axis=1)
# values的形状为(查询个数，“键－值”对个数)
return paddle.bmm(self.attention_weight.unsqueeze(1),
values.unsqueeze(-1)).reshape((-1, ))
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# X_tile的形状:(n_train，n_train)，每一行都包含着相同的训练输入
X_tile = np.tile(x_train, (n_train, 1))
# Y_tile的形状:(n_train，n_train)，每一行都包含着相同的训练输出
Y_tile = np.tile(y_train, (n_train, 1))
# keys的形状:('n_train'，'n_train'-1)
keys = X_tile[(1 - np.eye(n_train)).astype('bool')].reshape((n_train, -1))
# values的形状:('n_train'，'n_train'-1)
values = Y_tile[(1 - np.eye(n_train)).astype('bool')].reshape((n_train, -1))
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# X_tile的形状:(n_train，n_train)，每一行都包含着相同的训练输入
X_tile = x_train.repeat((n_train, 1))
# Y_tile的形状:(n_train，n_train)，每一行都包含着相同的训练输出
Y_tile = y_train.repeat((n_train, 1))
# keys的形状:('n_train'，'n_train'-1)
keys = X_tile[(1 - torch.eye(n_train)).type(torch.bool)].reshape((n_train, -1))
# values的形状:('n_train'，'n_train'-1)
values = Y_tile[(1 - torch.eye(n_train)).type(torch.bool)].reshape((n_train, -1))
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# X_tile的形状:(n_train，n_train)，每一行都包含着相同的训练输入
X_tile = tf.repeat(tf.expand_dims(x_train, axis=0), repeats=n_train, axis=0)
# Y_tile的形状:(n_train，n_train)，每一行都包含着相同的训练输出
Y_tile = tf.repeat(tf.expand_dims(y_train, axis=0), repeats=n_train, axis=0)
# keys的形状:('n_train'，'n_train'-1)
keys = tf.reshape(X_tile[tf.cast(1 - tf.eye(n_train), dtype=tf.bool)], shape=(n_train, -1))
# values的形状:('n_train'，'n_train'-1)
values = tf.reshape(Y_tile[tf.cast(1 - tf.eye(n_train), dtype=tf.bool)], shape=(n_train, -1))
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# X_tile的形状:(n_train，n_train)，每一行都包含着相同的训练输入
X_tile = x_train.tile([n_train, 1])
# Y_tile的形状:(n_train，n_train)，每一行都包含着相同的训练输出
Y_tile = y_train.tile([n_train, 1])
# keys的形状:('n_train'，'n_train'-1)
keys = X_tile[(1 - paddle.eye(n_train)).astype(paddle.bool)].reshape((n_train, -1))
# values的形状:('n_train'，'n_train'-1)
values = Y_tile[(1 - paddle.eye(n_train)).astype(paddle.bool)].reshape((n_train, -1))
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\diilbookstyleinputcell
.. code:: python
net = NWKernelRegression()
net.initialize()
loss = gluon.loss.L2Loss()
trainer = gluon.Trainer(net.collect_params(), 'sgd', {'learning_rate': 0.5})
animator = d2l.Animator(xlabel='epoch', ylabel='loss', xlim=[1, 5])
for epoch in range(5):
with autograd.record():
l = loss(net(x_train, keys, values), y_train)
l.backward()
trainer.step(1)
print(f'epoch {epoch + 1}, loss {float(l.sum()):.6f}')
animator.add(epoch + 1, float(l.sum()))
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\diilbookstyleinputcell
.. code:: python
net = NWKernelRegression()
loss = nn.MSELoss(reduction='none')
trainer = torch.optim.SGD(net.parameters(), lr=0.5)
animator = d2l.Animator(xlabel='epoch', ylabel='loss', xlim=[1, 5])
for epoch in range(5):
trainer.zero_grad()
l = loss(net(x_train, keys, values), y_train)
l.sum().backward()
trainer.step()
print(f'epoch {epoch + 1}, loss {float(l.sum()):.6f}')
animator.add(epoch + 1, float(l.sum()))
.. figure:: output_nadaraya-waston_736177_144_0.svg
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\diilbookstyleinputcell
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net = NWKernelRegression()
loss_object = tf.keras.losses.MeanSquaredError()
optimizer = tf.keras.optimizers.SGD(learning_rate=0.5)
animator = d2l.Animator(xlabel='epoch', ylabel='loss', xlim=[1, 5])
for epoch in range(5):
with tf.GradientTape() as t:
loss = loss_object(y_train, net(x_train, keys, values)) * len(y_train)
grads = t.gradient(loss, net.trainable_variables)
optimizer.apply_gradients(zip(grads, net.trainable_variables))
print(f'epoch {epoch + 1}, loss {float(loss):.6f}')
animator.add(epoch + 1, float(loss))
.. figure:: output_nadaraya-waston_736177_147_0.svg
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\diilbookstyleinputcell
.. code:: python
net = NWKernelRegression()
loss = nn.MSELoss(reduction='none')
trainer = paddle.optimizer.SGD(learning_rate=0.5, parameters=net.parameters())
animator = d2l.Animator(xlabel='epoch', ylabel='loss', xlim=[1, 5])
for epoch in range(5):
trainer.clear_grad()
l = loss(net(x_train, keys, values), y_train)
l.sum().backward()
trainer.step()
print(f'epoch {epoch + 1}, loss {float(l.sum()):.6f}')
animator.add(epoch + 1, float(l.sum()))
.. figure:: output_nadaraya-waston_736177_150_0.svg
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\diilbookstyleinputcell
.. code:: python
# keys的形状:(n_test，n_train)，每一行包含着相同的训练输入（例如，相同的键）
keys = np.tile(x_train, (n_test, 1))
# value的形状:(n_test，n_train)
values = np.tile(y_train, (n_test, 1))
y_hat = net(x_test, keys, values)
plot_kernel_reg(y_hat)
.. figure:: output_nadaraya-waston_736177_156_0.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
# keys的形状:(n_test，n_train)，每一行包含着相同的训练输入（例如，相同的键）
keys = x_train.repeat((n_test, 1))
# value的形状:(n_test，n_train)
values = y_train.repeat((n_test, 1))
y_hat = net(x_test, keys, values).unsqueeze(1).detach()
plot_kernel_reg(y_hat)
.. figure:: output_nadaraya-waston_736177_159_0.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
# keys的形状:(n_test，n_train)，每一行包含着相同的训练输入（例如，相同的键）
keys = tf.repeat(tf.expand_dims(x_train, axis=0), repeats=n_test, axis=0)
# value的形状:(n_test，n_train)
values = tf.repeat(tf.expand_dims(y_train, axis=0), repeats=n_test, axis=0)
y_hat = net(x_test, keys, values)
plot_kernel_reg(y_hat)
.. figure:: output_nadaraya-waston_736177_162_0.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
# keys的形状:(n_test，n_train)，每一行包含着相同的训练输入（例如，相同的键）
keys = x_train.tile([n_test, 1])
# value的形状:(n_test，n_train)
values = y_train.tile([n_test, 1])
y_hat = net(x_test, keys, values).unsqueeze(1).detach()
plot_kernel_reg(y_hat)
.. figure:: output_nadaraya-waston_736177_165_0.svg
.. raw:: html
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
d2l.show_heatmaps(np.expand_dims(
np.expand_dims(net.attention_weights, 0), 0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
.. figure:: output_nadaraya-waston_736177_171_0.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
d2l.show_heatmaps(net.attention_weights.unsqueeze(0).unsqueeze(0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
.. figure:: output_nadaraya-waston_736177_174_0.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
d2l.show_heatmaps(tf.expand_dims(
tf.expand_dims(net.attention_weights, axis=0), axis=0),
xlabel='Sorted training inputs',
ylabel='Sorted testing inputs')
.. figure:: output_nadaraya-waston_736177_177_0.svg
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
d2l.show_heatmaps(net.attention_weight.unsqueeze(0).unsqueeze(0),
xlabel='Sorted training inputs',
ylabel='Sorter testing, inputs')
.. figure:: output_nadaraya-waston_736177_180_0.svg
.. raw:: html
.. raw:: html
.. raw:: html
`Discussions `__
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`Discussions `__
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`Discussions `__
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